also be used to model how quantum information is transferred in a quantum field. Mathematical formulation 1 The Hamiltonian that describes the full system, consists of the free field Hamiltonian, the atomic excitation Hamiltonian, and the Jaynes–Cummings interaction Hamiltonian: Here, for convenience, the vacuum field energy is set to . For deriving the JCM interaction Hamiltonian the quantized radiation field is taken to consist of a single bosonic mode with the field operator , where the operators and are the bosonic creation and annihilation operators and is the angular frequency of the mode. On the other hand, the two-level atom is equivalent to a spin-half whose state can be described using a three-dimensional Bloch vector. (It should be understood that "two-level atom" here is not an actual atom with spin, but rather a generic two-level quantum system whose Hilbert space is isomorphic to a spin-half.) The atom is coupled to the field through its polarization operator . The operators and are the raising and lowering operators of the atom. The operator is the atomic inversion operator, and is the atomic transition frequency. Jaynes–Cummings Hamiltonian 1 Moving from the Schrödinger picture into the interaction picture (a.k.a. rotating frame) defined by the choice , we obtain This Hamiltonian contains both quickly and slowly oscillating components. To get a solvable model, the quickly oscillating "counter-rotating" terms, , are ignored. This is referred to as the rotating wave approximation, and it is valid since the fast oscillating term couples states of comparatively large energy difference: When the difference in energy is much larger than the coupling, the mixing of these states will be small, or put differently, the coupling is responsible for very little population transfer between the states. Transforming back into the Schrödinger picture the JCM Hamiltonian is thus written as Eigenstates It is possible, and often very helpful, to write the Hamiltonian of the full system as a sum of two commuting parts: where with called the detuning (frequency) between the field and the two-level system. The eigenstates of , being of tensor product form, are easily solved and denoted by , where denotes the number of radiation quanta in the mode. As the states and are degenerate with respect to for all , it is enough to diagonalize in the subspaces . The matrix elements of in this subspace, read For a given , the energy eigenvalues of are where is